Question

the following two - column proof proves the pythagorean theorem using similar triangles.

statement	justification
let \\(\overline{bc}=a\\), \\(\overline{ca}=b\\), \\(\overline{ab}=c\\), \\(\overline{cd}=h\\), \\(\overline{db}=y\\), \\(\overline{ad}=x\\)	by labeling
\\(y + x = c\\)	segment addition postulate
\\(\frac{c}{a}=\frac{a}{y}\\), \\(\frac{c}{b}=\frac{b}{x}\\)	?
\\(a^{2}=cy\\), \\(b^{2}=cx\\)	cross product property
\\(a^{2}+b^{2}=cy + b^{2}\\)	addition property of equality
\\(a^{2}+b^{2}=cy + cx\\)	substitution
\\(a^{2}+b^{2}=c(y + x)\\)	distributive property of equality
\\(a^{2}+b^{2}=c(c)\\)	substitution
\\(a^{2}+b^{2}=c^{2}\\)	multiplication

which of the following is the missing justification in the proof?

• substitution
• addition property of equality
• pieces of right triangles similarity theorem
• transitive property of equality

Explanation:

In the proof of the Pythagorean Theorem using similar triangles, the step with the proportion \(\frac{c}{a}=\frac{a}{y}\) and \(\frac{c}{b}=\frac{b}{x}\) relies on the Pieces of Right Triangles Similarity Theorem (also known as the Geometric Mean Theorem or Altitude-on-Hypotenuse Theorem). This theorem states that when an altitude is drawn to the hypotenuse of a right triangle, the two smaller right triangles are similar to the original triangle and to each other, leading to the proportional relationships seen here. The Cross Product Property is used after establishing these proportions, but the justification for the proportions themselves comes from the similarity of the right triangles, which is covered by this theorem.

Answer:

Pieces of Right Triangles Similarity Theorem